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Integration by Parts Rule

Maths

Integration by Parts Rule

Last updated date: 25th Apr 2024

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Introduction

Mathematics as a subject requires a good amount of practice and conceptual clarity. Students often find the subject of Math to be difficult when compared to other subjects. So, to solve all your problems we have presented and explained the topic of Math in the simplest language.

Let's start learning Math together and you no longer will have any fear about the subject. In this particular article, students will get to learn about the following concepts -

What is Integration?
Methods of Integration
What is the Ilate Rule?
The rule for solving Integration
Frequently asked questions

What is Integration?

In Mathematics, when general operations like addition operations cannot be performed, we use Integration to add values on a large scale.
There are different types of methods in Mathematics to integrate functions.
Integration and differentiation are also a pair of inverse functions similar to addition - subtraction, and multiplication-division.
The process of finding functions whose derivative is given is named antidifferentiation or Integration.

\[\int_b^a f(x) dx \] = value of the anti - derivative at upper limit b – the value of the same anti - derivative at lower limit a.

Here’s What Integration is!

If \[\frac{d}{dx} (F(x) = f(x)\]

then, \[\int f(x) dx = F(x) + c \]

The function F(x) is called anti-derivative or integral or primitive of the given function f(x) and c is known as the constant of Integration or the arbitrary constant.

The function f(x) is called the integrand and f(x)dx is known as the element of Integration.

Points to Remember

Since the integral of a function isn’t definite, therefore it is generally referred to as an indefinite integral.

We can never find the integral of a function at a point; we always find the integral of a given function in an interval.

The integral of a function is not unique; integrals of a function differ by numbers.

Types of Integration Math or the Integration Techniques

Here’s a list of Integration Methods –

Integration by Substitution
Integration by Parts Rule
Integration by Partial Fraction
Integration of Some particular fraction
Integration Using Trigonometric Identities

In this article we are going to discuss the Integration by Parts rule, Integration by Parts formula, Integration by Parts examples, and Integration by Parts examples and solutions.

Integration by Parts Rule

If the integrand function can be represented as a multiple of two or more functions, the Integration of any given function can be done by using the Integration by Parts rule.
Let us take an integrand function that is equal to u(x) v(x).
In Mathematics, Integration by parts basically uses the ILATE rule that helps to select the first function and second function in the Integration by Parts method.
Integration by Parts formula,

\[\int u(x).v(x) dx = u(x) \int v(x).dx – (u′(x) \int v(x).dx).dx\]

The Integration by Parts formula, can be further written as integral of the product of any two functions = (First function × Integral of the second function) – Integral of

(differentiation of the first function) × Integral of the second function

From the Integration by Parts formula discussed above,

u is the function u(x)
v is the function v(x)
u' is the derivative of the function u(x)

Ilate Rule

In Integration by Parts, we have learned when the product of two functions is given to us then we apply the required formula. The integral of the two functions is taken, by considering the left term as the first function and the second term as the second function. This method is called the Ilate rule. Suppose, we have to integrate xex, then we consider x as the first function and ex as the second function. So basically, the first function is chosen in such a way that the derivative of the function could be easily integrated. Usually, the preference order of this rule is based on some functions such as Inverse, Algebraic, Logarithm, Trigonometric, Exponent. This rule helps us to solve Integration by Parts examples using the Integration by Parts formula.

What is the ILATE rule?

ILATE is a rule which helps to decide which term should you differentiate first and which term should you integrate first.

I-Inverse
L-Logarithmic
Algebraic
T-Trigonometric
E-Exponential

The term which is closer to I is differentiated first and the term which is closer to E is integrated first.

Note

Integration by Parts rule is not applicable for functions such as \[\int \sqrt{x sin x dx}\].
We do not add any constant while finding the integral of the second function.
Usually, if any function is a power of x or a polynomial in x, then we take it as the first function. However, in cases where another function is an inverse trigonometric function or logarithmic function, then we take them as the first function.

Rules to be Followed for Solving Integration by Parts Examples:

So we followed these steps:

Choose u and v functions
Differentiate u: u'
Integrate v: ∫v dx
Put u, u' and put ∫v dx into the given formula: u∫v dx −∫u' (∫v dx) dx
Simplify and solve the Integration by Parts examples

In simpler words, to help you remember, the following ∫u v dx becomes:

(u integral v) - integral of (derivative u, integral v)

Standard Integrals in Integration

\[\int x^n dx\]	\[\frac{x^{n+1}}{n+1} + C\] where n≠ -1.
\[\int sinx dx \]	\[- cos x + C\]
\[\int cosx dx \]	\[sin x + C\]
\[\int sec^2x dx \]	\[tan x +C \]
\[\int cosec^2x dx \]	\[- cot x + C \]
\[\int secx tanx dx \]	\[sec x + C \]
\[\int cosecx cotx dx \]	\[- cosec x + C \]

Let’s understand better by solving Integration by Parts examples and solutions.

Questions to be Solved

Question 1. What is ∫x cos(x) dx ?

Answer : We have x multiplied by cos(x), so Integration by Parts is a good choice.

First choose which functions for u and v:

u = x
v = cos(x)

So now, we have obtained it in the format ∫u v dx and we can proceed:

Differentiate u: u' = x' = 1

Integrate the v part : ∫v dx = ∫cos(x) dx = sin(x)

Now we can put it together and we get the answer:

FAQs on Integration by Parts Rule

1. What is the application of Integration by part rule?

Whenever you read any topic of Math you should first try to know the application of the particular topic. This process will help you understand the concepts better and you would be able to make use of the concepts learned.

Getting to the point about the application of Integration by part rule, as in the field of calculus or Mathematical analysis, their application is very wide. It helps in car manufacturing and designing, to determine its center of mass, weight, and the central axis. It is also used to determine the safety and security of different roads. Other areas of application can be space engineering, designing, etc.

Watch the video lecture on the topic to get better clarity on the topic.

2. What is the difference between Integration and differentiation?

Differentiation means the rate of change of a function. For example - if you have a pool full of water, and suppose the water from the pool is getting replaced every second which is the rate of change. When more water gets replaced the rate of change is high and vice versa. This rate of change is called the differentiation which is represented as dV/dt.

On the other hand, Integration is the accumulation or sum of a function over a range. It is a process where we can sum up all the tiny contributions of the water accumulated in the pool and out of the pool, where the intervals between the summation are made smaller, we get closer to the true result.

3. Is there any other method to solve the problem of Integration apart from the basic Integration rule?

Integration is the anti process of differentiation and it is also the sum or addition of small Integration on the confines intervals. So, the alternative method of solving Integration would be the addition of small fin differentials. However, the application of this method is restricted to some problems, such as computational Mathematics and graph evaluation of experimental data, etc. to know more about differentiation and the relation between differentiation and Integration check out the related study materials on the website of Vedantu.

4. How do you know which Integration rule to be used to solve a particular problem?

There is no specific written rule which can guide you to pick particular rules to solve the problem. However, as the students start practicing questions after understanding the concept they gradually get the idea about which Integration rule to pick. So, the more you practice the question the stronger your intuition level will get.

Thus, it is your hard work and patience to learn the topic which will help you get all the questions right by applying the correct rules to it.

5. Does Vedantu provide sample questions related to the topic of Integration?

We understand the importance of practicing questions of Mathematics. So, to help the students learn faster and score good marks we provide a set of sample papers and a set of past year papers on the website of Vedantu. Visit the website and have access to all the practice questions. For a better result, students should first learn about the topic from the related article and the video lecture. After which they should learn and evaluate through solving the questions.

6. What is Integration by Parts Formula?

The method Integration by Parts is known to be a special method of integration that is often useful. We use it when two functions are multiplied together, but are also helpful in many other ways. Let us see the rule of integration by parts: ∫u v dx equals u∫v dx −∫u' (∫v dx) dx. u is the function u(x) is the formula for Integration by Parts.

7. Can you use Integration by Parts on any Integral?

Yes, we can use integration by parts to integrate any function. But the real problem is that we want integration by parts to be used instead of a substitution method for every function in integration. And some functions can only be integrated using the integration by parts method like ln(x).

8. What is the Product Rule of Integration?

The Product Rule of Integration enables you to integrate the product of two functions. For example, through a series of mathematical somersaults, you can turn the following equation into a formula that's useful for integrating.

9. When Should I use Integration by Parts?

Integration by parts is for functions that can be written as the product of another function and a third function's derivative. A good rule of thumb to follow would be to try the u-substitution first, and then if you cannot reformulate your function into the correct form, then you can try integration by parts.